Question

Queueing Theory

apply elementary queueing theory equations to compute statistics for the various scheduling systems

assume exponential inter-arrival and service time distributions

At the South Loop Oil 'n Lube, a new customer arrives for an oil change every 20 minutes, and is annoyed to find, on average, 3 customers ahead of him (including the one being serviced).

How much longer, typically, will it take for the current customer to finish his oil change?

After that, how much longer will the new customer have to wait to complete his own oil change?

Answer 1

Solution

This is a direct application of M/M/1 queue system.

Back-up Theory

An M/M/1 queue system is characterized by arrivals following Poisson pattern with average rate λ, [this is also the same as exponential arrival with average inter-arrival time = 1/ λ] service time following Exponential Distribution with average service time of (1/µ) [this is also the same as Poisson service with average service rate = µ] and single service channel.

Let n = number of customers in the system and m = number of customers in the queue.

[Trivially, n = m + number of customers under service.]

Let (λ/µ) = ρ

The steady-state probability of n customers in the system is given by P_n = ρⁿ(1 - ρ) …………(1)

The steady-state probability of no customers in the system is given by P₀ = (1 - ρ) …….……(2)

Average queue length = E(m) = (λ²)/{µ(µ - λ)} ………………………………………........…..…..(3)

Average number of customers in the system = E(n) = (λ)/(µ - λ)…………………………....…..(4)

Now to work out the solution,

‘At the South Loop Oil 'n Lube, a new customer arrives for an oil change every 20 minutes’ => λ = 3 [one per 20 minutes means 3 per hour] ……………............................……………… (5)

‘and is annoyed to find, on average, 3 customers ahead of him (including the one being serviced).’ => E(n) = 3 ………………………………………………...................................…………………. (6)

Part (a)

(5), (6) and (4) => 3 = 3/(µ - 3) or

µ = 4 per hour, which also implies 15 minutes per customer. …………………….....…………. (7)

So, vide (7), typically, it will take 15 minutes for the current customer to finish his oil change ANSWER

Part (b)

Once the above customer is serviced, there are now 3 customers, including the new arrival. So, the time the new customer will have to wait to complete his own oil change

= 3 x 5 = 45 minutes. ANSWER

DONE